Finite Group Databasebeta

← #193214

Subgroups — #193214

Order 512.

1252 conjugacy classes of subgroups.

The subgroup lattice has 559 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34206C211yesmaximal, characteristic
2#21959C211yesmaximal, characteristic, complemented
3#21401C211yesmaximal, characteristic
4#21170C211yesmaximal, characteristic, complemented
5C2×#1547C211yesmaximal, characteristic
6#8926C212yesmaximal, complemented
7#8925C212yesmaximal, complemented
8#8924C212yesmaximal, complemented
9#8921C212yesmaximal, complemented
10#7897C211yesmaximal, characteristic, complemented
11C2×#1279C211yesmaximal, characteristic, complemented
12C2×#532C2211yescharacteristic
13C2×#520—21no—
14C2×#428—21no—
15C2×#423C2211yescharacteristic
16C22×C4⋊C8C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18#2367—22no—
19#2366C2212yes—
20#2365C2212yes—
21#2354—21no—
22#2352—22no—
23D4×C4⋊C4—21no—
24#2325—22no—
25#2311—21no—
26C4×#121—22no—
27#2285C2211yescharacteristic
28#2285C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30#1990C2211yescharacteristic
31#1990C2211yescharacteristic
32#1957C2211yescharacteristic
33#1957C2211yescharacteristic
34#1949—22no—
35#1948—21no—
36#1948—21no—
37#1948C2212yes—
38#1948C2212yes—
39#1948C2212yes—
40#1947C2212yes—
41#1893—22no—
42#1807—22no—
43C4×#102—22no—
44#1736—21no—
45#1547C418yescomplemented
46C2×#332C2212yes—
47C2×#331C2212yes—
48#1279C2212yes—
49#1279C2212yes—
50#1279C2212yes—
51#1279C2212yes—
52C22×#41—21no—
53C22×M16—21no—
54C23×C8—21no—
55C23×C8—21no—
56#532—22no—
57#532—22no—
58#532C2×C414yes—
59#532Q814yes—
60C2×C4⋊Q8C2311yescharacteristic
61C2×#127C2311yescharacteristic
62C2×#124—21no—
63C2×#121—21no—
64C2×#121D412yes—
65#520—44no—
66#520—44no—
67C2×C4×Q8—22no—
68C2×C4×D4—24no—
69C2×C4×D4—22no—
70C2×C4×D4—22no—
71C2×C4×D4—21no—
72C2×C4×D4D412yes—
73C2×#117—22no—
74C2×#117—21no—
75C2×#117—21no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4D411yescharacteristic
78C22×C4⋊C4D411yescharacteristic
79C22×C42—21no—
80C22×C42C2311yescharacteristic
81#428—44no—
82C2×#107—22no—
83C2×#106—22no—
84C2×#106—22no—
85C2×#106—22no—
86#423—22no—
87#423C2×C414yescomplemented
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8C2×C414yescomplemented
92C2×C4⋊C8C2312yes—
93C2×C4⋊C8C2312yes—
94C2×#102—22no—
95C2×#102—22no—
96C2×C4×C8—22no—
97C2×C4×C8—22no—
98C2×C4×C8—21no—
99C2×C4×C8C2311yescharacteristic
100C2×C4×C8C2311yescharacteristic
101#391—22no—
102#390—22no—
103#390—22no—
104#389—22no—
105#388—24no—
106#388—24no—
107#387—24no—
108#386—24no—
109#386—24no—
110#385—24no—
111#384D411yescharacteristic
112#384D411yescharacteristic
113#384C2311yescharacteristic
114#384C2311yescharacteristic
115#382—21no—
116#382—21no—
117#380—22no—
118#380—22no—
119#380—21no—
120#380—21no—
121#380—21no—
122#380D412yes—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4D412yes—
130C4×C4⋊C4C2312yes—
131C4×C4⋊C4C2312yes—
132C4×C22⋊C4—24no—
133C2×#95—21no—
134#340—22no—
135#340—22no—
136#340—22no—
137#340—22no—
138#340—22no—
139#340—22no—
140#332C2×C418yescomplemented
141#331C2×C418yescomplemented
142C2×#41—24no—
143C2×#41—24no—
144C2×#41—42no—
145C2×#41—42no—
146C2×#41—42no—
147C2×#41—42no—
148C22×Q8—21no—
149C22×D4—22no—
150C22×D4—21no—
151C22×D4—21no—
152C22×D4C2×D412yes—
153C23×C4—21no—
154C23×C4—22no—
155C23×C4—24no—
156C23×C4—21no—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C2×M16—24no—
161C2×M16—42no—
162C2×M16—42no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—24no—
168C22×C8—42no—
169C22×C8—42no—
170C22×C8—42no—
171C22×C8—81no—
172C22×C8—81no—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175C4⋊Q8C22×C412yes—
176C4⋊Q8C2×Q812yes—
177#127—21no—
178#127—21no—
179#127C22×C412yes—
180#127C2×Q812yes—
181#124—24no—
182#124—24no—
183#121—24no—
184#121—24no—
185#121—24no—
186#121C4⋊C418yes—
187C4×Q8—44no—
188C4×Q8—44no—
189C4×D4—24no—
190C4×D4—24no—
191C4×D4—24no—
192C4×D4—44no—
193C4×D4—416no—
194C4×D4—44no—
195C4×D4—82no—
196C4×D4—82no—
197C4×D4C4⋊C418yes—
198#117—48no—
199#117—42no—
200#117—42no—
201#117—42no—
202#117—42no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—48no—
221C2×C4⋊C4SD1614yes—
222C2×C4⋊C4C2×D412yes—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4C2×D412yes—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—41no—
234C2×C42—24no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—41no—
239C2×C42—81no—
240C2×C42—81no—
241C2×C42C22×C412yes—
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42C2×Q812yes—
246C2×C42#4112yes—
247C2×C42#4112yes—
248C2×C42C2411yescharacteristic
249#107—44no—
250#106—44no—
251#106—82no—
252#106—82no—
253C4⋊C8—24no—
254C4⋊C8—24no—
255C4⋊C8—22no—
256C4⋊C8—22no—
257C4⋊C8C22×C414yes—
258C4⋊C8C22×C414yes—
259#102—44no—
260#102—44no—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8—21no—
264C4×C8—21no—
265C4×C8—42no—
266#95—24no—
267#95—24no—
268#95—22no—
269#95—22no—
270#95—22no—
271#95—22no—
272#95—41no—
273#95—41no—
274C24—22no—
275C24#12011yescharacteristic
276#41—48no—
277#41—48no—
278#41—48no—
279#41—48no—
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×D4—28no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—44no—
295C2×D4—44no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4C2×C4⋊C414yes—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—28no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—48no—
319C22×C4—44no—
320C22×C4—44no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—81no—
325C22×C4—81no—
326C22×C4—81no—
327C22×C4—81no—
328C22×C4—84no—
329C22×C4—84no—
330C22×C4C2×C4⋊C412yes—
331C22×C4C4×D412yes—
332C22×C4C4×D412yes—
333C22×C4#12011yescharacteristic
334C22×C4#12011yescharacteristic
335C22×C4#12011yescharacteristic
336C22×C4C22⋊Q812yes—
337C22×C4C22⋊Q812yes—
338C22×C4#12312yes—
339C22×C4C2×SD1612yes—
340C22×C4#13512yes—
341C22×C4AGL(1,ℤ/8)12yes—
342C22×C4AGL(1,ℤ/8)12yes—
343C22×C4C22×D411yescharacteristic
344C22×C4C22×D411yescharacteristic
345C22×C4C22×D411yescharacteristic
346M16—48no—
347C2×C8—22no—
348C2×C8—22no—
349C2×C8—24no—
350C2×C8—24no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—48no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—84no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—24no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—48no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—81no—
387C4⋊C4—81no—
388C4⋊C4C2×C4⋊C414yes—
389C4⋊C4C2×SD1614yes—
390C22⋊C4—28no—
391C22⋊C4—28no—
392C22⋊C4—48no—
393C22⋊C4—44no—
394C22⋊C4—84no—
395C22⋊C4—84no—
396C22⋊C4—82no—
397C42—48no—
398C42—42no—
399C42—42no—
400C42—42no—
401C42—42no—
402C42—82no—
403C42—82no—
404C42C2×C4⋊C412yes—
405C42C4×D412yes—
406C42C4×D412yes—
407C42C22⋊Q812yes—
408C42C22⋊Q812yes—
409C42#12312yes—
410C42C23×C411yescharacteristic
411C42C22×Q811yescharacteristic
412C42C2×#4111yescharacteristic
413C42C2×#4111yescharacteristic
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—24no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23#38512yes—
426C23#44912yes—
427C23#45012yes—
428C23C2×#12011yescharacteristic
429Q8—42no—
430Q8—42no—
431Q8—42no—
432Q8—42no—
433D4—44no—
434D4—416no—
435D4—42no—
436D4—42no—
437D4—42no—
438D4—42no—
439D4—42no—
440D4—42no—
441D4—42no—
442D4—42no—
443D4—82no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—416no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—84no—
473C2×C4—84no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—168no—
481C2×C4#38512yes—
482C2×C4#38512yes—
483C2×C4#38512yes—
484C2×C4C4×SD1614yes—
485C2×C4#44214yes—
486C2×C4#44912yes—
487C2×C4#45012yes—
488C2×C4#47614yes—
489C2×C4#47814yes—
490C2×C4C22×C4⋊C411yescharacteristic
491C2×C4C2×C4×D411yescharacteristic
492C2×C4C2×C4×D411yescharacteristic
493C2×C4C2×C22⋊Q811yescharacteristic
494C2×C4C2×C22⋊Q811yescharacteristic
495C2×C4C2×#12311yescharacteristic
496C2×C4C22×SD1611yescharacteristic
497C2×C4C2×#13511yescharacteristic
498C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
499C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
500C2×C4#57611yescharacteristic
501C2×C4#57711yescharacteristic
502C2×C4#57711yescharacteristic
503C2×C4#57811yescharacteristic
504C8—42no—
505C8—42no—
506C8—44no—
507C8—44no—
508C22—22no—
509C22—22no—
510C22—22no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—44no—
515C22—44no—
516C22—44no—
517C22—82no—
518C22—82no—
519C22—82no—
520C22—82no—
521C22#186414yes—
522C22C2×#38511yescharacteristic
523C22C2×#45011yescharacteristic
524C22C2×#44911yescharacteristic
525C22#301211yescharacteristic
526C22#301311yescharacteristic
527C22#301811yescharacteristic
528C22#301811yescharacteristic
529C4—21no—
530C4—21no—
531C4—22no—
532C4—22no—
533C4—44no—
534C4—44no—
535C4—82no—
536C4—82no—
537C4—82no—
538C4—161no—
539C4—161no—
540C4#186414yes—
541C4C2×C4×SD1611yescharacteristic
542C4C2×#44211yescharacteristic
543C4#294611yescharacteristic
544C4#294711yescharacteristic
545C4C2×#47611yescharacteristic
546C4C2×#47811yescharacteristic
547C4#308311yescharacteristic
548C4#308411yescharacteristic
549C2—24no—
550C2—84no—
551C2#1866311yescharacteristic
552C2#1866611yescharacteristic
553C2#1867511yescharacteristic
554C2#1867811yescharacteristic
555C2#2085511yescharacteristic
556C2C2×#186411yescharacteristic
557C2#2087811yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #21959C2semidirect
4. #21170C2semidirect
6. #8926C2semidirect
7. #8925C2semidirect
8. #8924C2semidirect
9. #8921C2semidirect
10. #7897C2semidirect
11. C2×#1279C2semidirect
45. #1547C4semidirect
87. #423C2×C4semidirect
91. C2×C4⋊C8C2×C4semidirect
140. #332C2×C4semidirect
141. #331C2×C4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#342062
#219592
#214012
#211702
C2×#15472
#89262
#89252
#89242
#89212
#78972
C2×#12792